With the trivial euclidean metric \(d(x,y):=\Vert x-y\Vert \) , the transition kernel from (REF ) satisfies \(\mathcal {P}(x,\cdot ) \in T_1^{d}(1), \hspace{5.0pt} \forall x \in \mathcal {X}\)  [1]} and \(\mathcal {W}_{d}^{2}(\mathcal {P}(x,\cdot ),\mathcal {P}(y,\cdot )) =\Vert Ax-Ay \Vert  \le \hat{\lambda }d(x,y)\)  (see e.g., [2]}).
Now, an application of Jensens' inequality reveals \(\mathcal {W}_{d}(\mathcal {P}(x,\cdot ),\mathcal {P}(y,\cdot )) \le \mathcal {W}_{d}^{2}(\mathcal {P}(x,\cdot ),\mathcal {P}(y,\cdot ))\)  and contractivity follows. As conditions (REF ) and () are satisfied, we can use coupling technique inspired by [3]} to prove that \(\mathbb {P}^{N}:=\) Law\((y_1,\ldots , y_N)\)  of the LDS satisfies \(T_{1}^{d_{(N)}}(\frac{N}{(1-\hat{\lambda })^{2}})\) ; and if \(\nu _{\pi }\)  is the invariant measure corresponding to (REF ), it follows that
\(& \nonumber \mathbb {P}^{N} \Bigg [\bigg |\frac{1}{N} \sum _{i=1}^{N} \Vert y_i\Vert  - <\Vert y\Vert >_{\nu _{\pi }} \bigg | > \frac{\mathcal {W}_{d} (\mathcal {P}(x. \cdot ), \nu _{\pi })}{N( 1-\hat{\lambda })}+ \epsilon \Bigg ] \le \\ &  \le \mathbb {P}^{N} \Bigg [\bigg |\frac{1}{N} \sum _{i=1}^{N} \Vert y_i\Vert  -\mathbb {E}\Bigg (\frac{1}{N} \sum _{i=1}^{N} \Vert y_i\Vert  \Bigg ) \bigg | > \epsilon \Bigg ] \le 2\exp \bigg (-\frac{N \epsilon ^2 (1-\hat{\lambda })^2}{2}\bigg ).\) 
